Exercises 1. Write out the first 5 terms of the following sequences and describe the sequence using the words convergent, divergent, oscillating, periodic as appropriate (b) 2. What value does the sequence given by (a) (c) Ans: Divergent Ans:DivergentPeriodic Ans: ConvergentOscillating

Sum of Terms (S n ) u 1 = first termu n = last termd = common difference Arithmetic Series = ½ n (u 1 + u n ) Arithmetic Series = ½ n (2u 1 + (n – 1) d ) Why are both of these formulae useful? Mathsnet sum of n terms

Arithmetic Sequences and Series Consolidation Understand and calculate series defined using ∑ notation Solve more difficult problems involving Arithmetic Series Complete Tarsia Puzzle in groups

Series When the terms of a sequence are added, we get a series The sequence gives the series Sigma Notation for a Series A series can be described using the general term e.g. can be written is the Greek capital letter S, used for Sum 1 st value of n last value of n

Sigma Notation Sigma notation is a method of describing the sums of the terms of a sequence it follows this convention... The sigma symbol. Used to tell us that we are summing. Value of r for last term Value of r to start with rth term rule of the sequence being added.

How it works... The notation  tells us that we must sum the terms of the sequence with rth term rule r 2 for values of r from 3 to 6 so the result is = 86

Some Observations The sum of (a constant times a sequence) gives that constant times the sum of the sequence. The sum of (the sum of two sequences) has the same sum as the sum of the two sequences summed individually.

Oddities BODMAS can be some what confusing. Powers, multiplication and division occur before sigma. Addition and subtraction should be placed in brackets (although this does not seem to be a hard and fast rule!)

rth term rules for +,-,+,- sequences A sequence that alternates between positive and negative (and will therefore add some terms and take others) will involve multiplying by a negative. 2,4,6,8,10... Has rth term u r =2r 2,-4,6,-8,10... Has rth term u r =(2r)(-1) r-1 The (-1) r-1 gives the changing signs 2 nd,4 th etc (-1) r will change the opposite terms 1 st, 3 rd etc

Examples... Write a rule in sigma notation for these sums Calculate this sum.

(a) (b) 2. Write the following using sigma notation Exercises 1. Write out the first 3 terms and the last term of the series given below in sigma notation (a) (b) n = 1 n = 2 n = 20

Group work In pairs attempt 8 piece rhombus puzzle 3 pairs work together to complete 24 piece hexagon puzzle Reward to winning 6

Exercises 1. The 1 st term of an A.P. is 20 and the sum of 16 terms is 280. Find the last term and the common difference. 2. Solution: Find the sum of the series given by We can see the series is arithmetic so, Substituting n = 1, 2 and 3, we get  6,  2, 2